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7 Nov 2012 Deformation Theory and Rational Homotopy Type 1 Deformation Theory and Rational Homotopy Type 2 Mike Schlessinger and Jim Stasheff 3 November 8, 2012 4 Abstract 5 We regard the classification of rational homotopy types as a problem in algebraic de- 6 formation theory: any space with given cohomology is a perturbation, or deformation, 7 of the “formal” space with that cohomology. The classifying space is then a “moduli” 8 space — a certain quotient of an algebraic variety of perturbations. The description 9 we give of this moduli space links it with corresponding structures in homotopy theory, p 10 especially the classification of fibres spaces F → E → B with fixed fibre F in terms 11 of homotopy classes of maps of the base B into a classifying space constructed from 12 Aut(F ), the monoid of homotopy equivalences of F to itself. We adopt the philosophy, 13 later promoted by Deligne in response to Goldman and Millson, that any problem in 14 deformation theory is “controlled” by a differential graded Lie algebra, unique up to 15 homology equivalence (quasi-isomorphism) of dg Lie algebras. Here we extend this 16 philosophy further to control by L∞ -algebras. In memory of Dan Quillen who established the foundation on which this work rests. 17 Contents arXiv:1211.1647v1 [math.QA] 7 Nov 2012 18 1 Introduction 3 19 1.1 Background .................................... 3 20 1.2 ControlbyDGLAs ................................ 4 21 1.3 Applications.................................... 7 22 1.4 Outline....................................... 8 23 2 Models of homotopy types 10 24 2.1 The Tate–Jozefiak resolution in characteristic zero . ..... 10 25 2.2 TheHalperin–Stashefforfilteredmodel . 11 26 3 Differential graded Lie algebras, models and perturbations 14 27 3.1 Differential graded commutative coalgebras and dg Lie algebras . ..... 15 28 3.2 ThetensoralgebraandfreeLiealgebra . 15 29 3.3 ThetensorcoalgebraandfreeLiecoalgebra . ... 17 1 Deformation Theory and Rational Homotopy Type November 8, 2012 2 30 3.4 The standard construction C(L) ........................ 18 31 3.5 The Quillen and Milnor/Moore et al spectral sequences [51, 44, 45]..... 19 32 3.6 The standard construction A(L) ........................ 20 33 3.7 Comparison of Der L and Der C(L) ...................... 21 34 3.8 A(L(H))andfilteredmodels .......................... 22 35 4 Classifying maps of perturbations and homotopies: The Main Homotopy 36 Theorem. 25 37 4.1 Homotopyofcoalgebramaps .......................... 25 38 4.2 ProofoftheMainHomotopyTheorem . 28 39 5 Homotopy invariance of the space of homotopy types 31 40 6 Control by L∞-algebras. 35 41 6.1 Quasi-isomorphisms and homotopy inverses . ... 35 42 6.2 L∞-structure on H(L) .............................. 36 43 7 The Miniversal Deformation 38 44 7.1 Introduction.................................... 38 45 7.2 Varietiesandschemes............................... 39 46 7.3 Versaldeformations................................ 40 47 7.4 Theminiversaldeformation ........................... 41 48 7.5 Gauge equivalence for nilpotent L∞-algebras.................. 44 49 7.6 Summary ..................................... 45 50 8 Examples and computations 47 51 8.1 Shallowspaces................................... 47 52 8.2 CellstructuresandMasseyproducts . ... 48 53 8.3 Moderatelyshallowspaces............................ 49 54 8.4 Moremoderatelyshallowspaces . .. .. 50 55 8.5 Obstructions ................................... 51 56 8.6 Morecomplicatedobstructions. 55 57 8.7 Othercomputations................................ 56 58 9 Classification of rational fibre spaces 57 59 9.1 Algebraicmodelofafibration .......................... 57 60 9.2 Classificationtheorem .............................. 59 61 9.3 Examples ..................................... 61 62 9.4 Openquestions.................................. 64 63 10 Postscript 64 Deformation Theory and Rational Homotopy Type November 8, 2012 3 64 1 Introduction 65 In this paper, we regard the classification of rational homotopy types as a problem in algebraic 66 deformation theory: any space with given cohomology is a perturbation, or deformation, of 67 the “formal” space with that cohomology. The classifying space is then a “moduli” space 68 — a certain quotient of an algebraic variety of perturbations. The description we give of 69 this moduli space links it with others which occur in algebra and topology, for example, 70 the moduli spaces of algebras or complex manifolds. On the other hand, our dual vision 71 emphasizes the analogy with corresponding structures in homotopy theory, especially the p 72 classification of fibres spaces F → E → B with fixed fibre F in terms of homotopy classes 73 of maps of the base B into a classifying space constructed from Aut(F ), the monoid of 74 homotopy equivalences of F to itself. In particular, the moduli space of rational homotopy 75 types with fixed cohomology algebra can be identified with the space of “path components” 76 of a certain differential graded coalgebra. 77 Although the majority of this paper is concerned with constructing and verifying the 78 relevant machinery, the final sections are devoted to a variety of examples, which should 79 be accessible without much of the machinery and might provide motivation for reading the 80 technical details of the earlier sections. 81 Portions of our work first appeared in print in [56, 57] and then in ‘samizdat’ versions over 82 the intervening decades (!), partly due to some consequences of the mixture of languages. 83 Some of those versions have worked there way into work of other researchers; we have tried 84 to maintain much of the flavor of our early work while taking advantage of progress made 85 by others. 86 Crucially, throughout this paper, the ground field is the rational numbers, Q (character- 87 istic 0 is really the relevant algebraic fact), although parts of it make sense even over the 88 integers. 89 1.1 Background 90 Rational homotopy theory regards rational homotopy equivalence of two simply con- 91 nected spaces as the equivalence relation generated by the existence of a map f : X → Y ∗ ∗ ∗ 92 inducing an isomorphism f : H (Y ; Q) → H (X, Q). Here we are much closer to a complete 93 classification than in the ordinary (integral) homotopy category. An obvious invariant is the ∗ 94 cohomology algebra H (X; Q). Halperin and Stasheff [20] showed that all simply connected 95 spaces X with fixed cohomology algebra H of finite type over Q can be described (up to 96 rational homotopy type) as follows: (Henceforth ‘space’ shall mean ‘simply connected space 97 of finite type’ unless otherwise specified.) 98 Resolve H by a d(ifferential) g(raded) c(ommutative) a(lgebra) (SZ,d) which is con- 99 nected and free as a graded commutative algebra with a map (SZ,d) → H of dgcas inducing 100 H(SZ,d) ≃ H. Here S denotes graded symmetric algebra. (See section 2.1 for details, espe- 101 cially in re: the various gradings involved.) The notation Λ instead of S is often used within 102 rational homotopy theory, where it is a historical accident derived from de Rham theory. Deformation Theory and Rational Homotopy Type November 8, 2012 4 ∗ 103 Let A (X) denote a differential graded commutative algebra of “differential forms over the 104 rationals” for the space X, e.g. Sullivan’s version of the deRham complex [63, 6]). Given ≃ ∗ 105 an isomorphism i : H → H (X), there is a perturbation p (a derivation of SZ of degree 2 106 1 which lowers resolution degree by at least 2 such that (d + p) = 0) and a map of dga’s ∗ 107 (SZ,d + p) → A (X) inducing an isomorphism of rational cohomology. If X and Y have 108 the same rational homotopy type, the perturbations pX and pY must be related in a certain 109 way, spelled out in §2.2. This is one of several ways (cf. [36, 13]) it can be seen that 110 Main Theorem 1.1. For fixed H, the set of homotopy types of pairs (X, i : H ≃ H(X)) 111 can be represented as the quotient V/G of a (perhaps infinite dimensional) conical rational 112 algebraic variety V modulo a pro-unipotent (algebraic) group action G. 113 Corollary 1.2. The set of rational homotopy types with fixed cohomology H can be repre- 114 sented as a quotient Aut H\V/G. 115 1.2 Control by DGLAs 116 The variety and the group can be expressed in the following terms: let Der SZ denote 117 the graded Lie algebra of graded derivations of SZ, which is itself a dg Lie algebra (see 118 Definition 3.1) with the commutator bracket and the differential induced by the internal 119 differential on SZ. Let L ⊂ DerSZ be the sub–Lie algebra of derivations that decrease 1 120 the sum of the total degree plus the resolution degree. The variety V ⊂ L is precisely 1 2 121 V = {p ∈ L |(d + p) =0}. In fact, V is the cone on a projective variety (of possibly infinite 122 dimension) §2.3 The pro-unipotent group G is exp L, which acts via the adjoint action of L 123 on d + p. 124 We said above that we regard our problem as one of deformation theory in the (homolog- 125 ical) algebra sense. A commutative algebra H has a Tate resolution which is an almost free 126 commutative dga SZ, that is, free as graded commutative algebra, ignoring the differential. 127 A deformation of H corresponds to a change in differential d → d + p on SZ. Instead of 128 L ⊂ DerSZ as above, the sub-dg Lie algebra L¯ ⊂ DerSZ of nonpositive resolution degree 129 is used. 130 As far as we know, deformation theory arose with work on families of complex structures. 131 Early on, these were expressed in terms of a moduli space [53, 69]. This began with Riemann, 132 who first introduced the term ”moduli”. He proved that the number of moduli of a surface 133 of genus 0 was 0. 1 for genus 1 and 3g − 3 for g > 1. These are the same as the numbers 134 of quadratic differentials on the surface, which was perhaps the impetus for Teichmueller’s 135 work identifying these as ”infinitesimal deformations”.
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